Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.

Algebraic structure → Group theory
Group theory
Basic notions
Group homomorphisms
Finite groups
Classification of finite simple groups
Modular groups
  • PSL(2, )
  • SL(2, )
Topological and Lie groups
Infinite dimensional Lie group
  • O(∞)
  • SU(∞)
  • Sp(∞)
Algebraic groups

The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004.

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